All I can say is that I hope for your sake that the shop is honest. If the stator is still mounted in the engine, then it isn't very difficult for YOU to follow my simple instructions and conduct the tests. The alternative is to remove the stator and take it into that shop. In order to remove the stator, you have to dismantle the PTO clutch on the front and then obtain a flywheel puller to pop the flywheel loose. I have to wonder if your shadetree mechanic abilities extend that far, no insult implied.
More often than not, the true fault is not with the actual stator but with other areas of the electrical system such as the voltage regulator, bad connections, bad grounds, corroded wiring, broken wires or improper wiring. It's your life and your money but no matter how you cut it, you are either going to have to dive into this tractor, find and correct the charging problem or.....haul the tractor to a shop and pay them to find it for you. That could get very expensive to the point of it exceeding the current retail value of your 444. My way is for you to go and purchase a $20.00 multimeter and work with me to get to the real problem. Throwing new parts at a problem is an absurd way to correct it.
As a young man, I worked in a gas station that practiced "parts replacement" as a diagnostic tool. A customer was constantly plagued with a car that would not start due to a dead battery. Over the course of a month, the stupid gas station owner sold and installed a new battery, new starter motor, new alternator, new regulator and new battery cables, all to no avail. When he finally ran out of electrical parts that he could change, one of the better mechanics began to really search for the cause which turned out to be a loose nut on the back of the ammeter. The nut was tightened and the problem went away. In the meantime, the customer's perfectly good parts were no longer available to be re-installed and the money he spent was also gone. I learned a lot from observing that experience as well as a few similar ones at that location.
A multimeter is easy to use. The initial test will take less than 5 minutes of your time. Send me a PM with your phone number and I will guide you through it.